Abstract
In this paper, we consider the heat conduction operator with elliptic part of divergent type on a stratified set (i.e., on the set of manifolds of various dimension). We prove an analog of the lemma on the normal derivative and the weak and strong maximum principles for parabolic inequalities on this set.
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REFERENCES
V. V. Kulyaba, “The problem of heat conduction leading to a parabolic operator on a stratified set,” in: Collection of Papers of the Young Scientists of Faculty of Mathematics of Voronezh State University [in Russian], Voronezh, 2001, pp. 109–112.
G. Lumer, “Espaces ramifiés et diffusions sur les réseaux topologiques,” C. R. Acad. Sc. Paris. Sér. A, 291 (1980), 627–630.
G. Lumer, “Connecting of local operators and evolution equations on networks,” in: Lecture Notes in Math., vol. 787, Springer-Verlag, New York, 1980, pp. 219–234.
Yu. V. Pokornyi, “On boundary-value problems on graphs,” in: Numerical Methods and Optimization. Proceedings of IV Simposium of Estonian Academy of Sciences [in Russian], Tallin, 1988, pp. 158–161.
S. Nicaise, “Spectre des réseaux topologiques finis,” Bull. Sc. Math. Ser. 2, 111 (1987), 401–413.
S. Nicaise, “Le laplacien sur les réseaux deux-dimensionnels polygonaux topologiques,” J. Math. Pures Appl., 67 (1988), 93–113.
A. Gavrilov, S. Nicaise, and O. Penkin, Poincaré's Inequality on Stratified Sets and Applications, Rapport de recherche no. 01.2 (Février 2001), Université de Valenciennes, 2001.
V. V. Zhikov, “Connectedness and averaging. Examples of fractal conductivity,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 187 (1996), no. 8, 3–40.
V. V. Zhikov, “On an extension and application of the method of two-scaled convergence,” Mat. Sb. [Russian Acad. Sci. Sb. Math.], 191 (2000), no. 7, 31–72.
O. M. Penkin and Yu. V. Pokornyi, “On incompatible inequalities for elliptic operators on stratified sets,” Differentsial'nye Uravneniya [Differential Equations], 34 (1998), no. 8, 1107–1113.
D. Gilbarg and M. N. Trudinger, Elliptic Partial Differential Equations of Second Order [in Russian], Nauka, Moscow, 1989.
O. Penkin, “About a geometrical approach to multistructures and some qualitative properties of solutions,” in: Partial Differential Equations on Multistructures (F. Ali Mehmeti, J. von Belov, and S. Nicaise, editors), Dekker, Marcel, 2001, pp. 183–191.
A. A. Gavrilov and O. M. Penkin, “An analog of the lemma on normal derivative for an elliptic equation on a stratified set,” Differentsial'nye Uravneniya [Differential Equations], 36 (2000), no. 2, 226–232.
I. Rubinstein and L. Rubinstein, Partial Differential Equations in Classical Mathematical Physics, Cambridge Univ. Press, Cambridge, 1998.
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Kulyaba, V.V., Penkin, O.M. The Maximum Principle for Parabolic Inequalities on Stratified Sets. Mathematical Notes 73, 228–239 (2003). https://doi.org/10.1023/A:1022163109895
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DOI: https://doi.org/10.1023/A:1022163109895